Chi-square oracle inequalities

By Iain M. Johnstone. 2000

75 citations recorded by the source corpus.

Subfields: Computer Vision and Pattern Recognition, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/85a72aba2ffc9e3b46fff5b0ef498ce56a3a29d1

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In Computer Vision and Pattern Recognition

In Finance

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Citations in the corpus

On Consistency and Sparsity for Principal Components Analysis in High Dimensions (2009, unknown): “logð1 þ Þ # =2 1 < < 0; 3 =8 0 # < 12 ; (the latter following, for example, from (47) in Johnstone (2001)) yield”
Compressed and Privacy-Sensitive Sparse Regression (2009, background): “The proof of this result uses inequalities from [31] and [32] on random variables and the sum of products of normals.”
ADAPTIVE ESTIMATION OF A QUADRATIC FUNCTIONAL BY MODEL SELECTION (2001, methodology): “Note that a similar estimator has been introduced ﬁrst by Gayraud and Tribouley (1999), the difference being that in their procedure one considers as above or in Johnstone (1999).”
Sparse Principal Components Analysis (2009, methodology): “valid for n ≥ 16 and 0 ≤ t ≤ n1/6 (Johnstone 2001).”
Model selection via testing: an alternative to (penalized) maximum likelihood estimators (2006, unknown): “Alternatively, when βȳ2 " q/2, we can use the bound (Inequality 45 from Johnstone [36]) ∫ +∞ z x t e−x dx < (zt+1 e−z)(z − t)−1 for z > t , which gives ∫ ∞ βȳ2 x q/2−1 e−x dx < (βȳ2)q/2 exp(−βȳ2) so that finally E[Yq ] − ȳq αȳq ! { √ π eq/2 [ q/(2 eβȳ2) ]q/2 for all ȳ > 0, (q/2) exp(−βȳ2) if βȳ2 " q”
Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using $\ell _{1}$ -Constrained Quadratic Programming (Lasso) (2009, background): “The bound (54a) is taken from Johnstone [21], whereas the bound (54b) follows from Laurent and Massart [23].”
Prediction by Supervised Principal Components (2006, unknown): “Johnstone (2001) showed that P ( χ2(n) > n(1 + ) ) ≤ e−3n 2/16, 0 < < 1 2 , (A.6) P ( χ2(n) < n(1 − ) ) ≤ e−n 2/4, 0 < < 1, (A.7) and P ( χ2(n) > n(1 + ) ) ≤ √ 2 √ n e−n 2/4 0 n1/16,n ≥ 16.”
Minimal Penalties for Gaussian Model Selection (2007, background): “The proof of this proposition is mainly based on the following elementary lemma which controls the tails of gamma integrals (see Johnstone, [23], proof of Lemma 6.1).”
Nonquadratic estimators of a quadratic functional (2005, background): “2, 3 and 5 hold for Q˜(m) with the same choices of m. Finally we should also note that the term-by-term thresholding procedures used here are quite diﬀerent from the global thresholding rules used in [21] and [27], which are designed for estimation over quadratically convex parameter spaces where th”
High-dimensional analysis of semidefinite relaxations for sparse principal components (2008, unknown): “ −d≥2 √ (60a) dx+2x] ≤exp(−x) and P[X −d≤−2 √ (60b) dx]≤exp(−x). We also use the following slightly diﬀerent version of the bound (60a), P[X −d≥dx]≤exp(− 3 16dx 2), 0≤x< 1 (61) 2, due to Johnstone [19]. More generally, the analogous tail bounds for noncentral χ2, taken from Birg´e [4], can be est”